Problem A. 879. (April 2024)

A. 879. An integer \(\displaystyle k>2\) is given. Xavier and Yvette play the following game: a number \(\displaystyle n>k\) is initially written on the blackboard. The two players take turns, Xavier starts. In each turn the integer \(\displaystyle m\) on the blackboard is replaced by integer \(\displaystyle m'\) satisfying \(\displaystyle k\le m'<m\) and \(\displaystyle \gcd (m,m')=1\). The player who cannot make a legal move loses the game. We say that integer \(\displaystyle n>k\) is good if Yvette has a winning strategy. Prove that if \(\displaystyle n\), \(\displaystyle n'>k\) are two integers satisfying the condition that every prime \(\displaystyle p\le k\) divides \(\displaystyle n\) if and only if it divides \(\displaystyle n'\), then \(\displaystyle n\) is good if and only if \(\displaystyle n'\) is good.

(7 pont)

Deadline expired on May 10, 2024.

Unfortunately this problem can be found on the 2013 IMO shortlist (this was not intended). The official solution can be found here:

Official Solution

Statistics:

8 students sent a solution.
7 points:	Bodor Mátyás, Philip Stefanov, Varga Boldizsár, Wiener Anna.
6 points:	Szakács Ábel.
5 points:	1 student.
1 point:	1 student.
0 point:	1 student.

Problems in Mathematics of KöMaL, April 2024